A primitive cell is the smallest cell you can use to build up your crystal by simple repetition. The choice is not unique. In fact there is an infinite number of possible choices, but they turn out to all have the same volume.

In some cases like FCC or BCC one often uses a "conventional" unit cell that is bigger than a primitive one. This is done because the conventional unit cell more intuitively shows the symmetry of the crystal.

In the end, if you do your math correctly, the choice does not matter.

Primitive unit cell contains only one lattice point while conventional unit cell have more than one point, and as said by M Quack are usually considered for symmetry purpose.

That is how is defined in my book. Thus, the disadvantage of using a primitve cell is that how you can only have one lattice point inside of, sometimes you will get a pretty weird shaped cell. If you are more flexible, and if you decide to have more lattice points inside of your unit cell, you can get a better shape. For instance, if you use a cube with for lattice points, you can describe the Bravais lattice of a FCC very nicely. However, if you try to describe this very same lattice with a primitive cell, you will get a very weird thing.
Please, let me know if that helped you. I am also studying this subject, so we could discuss it together in more depth.

That is how is defined in my book. Thus, the disadvantage of using a primitve cell is that how you can only have one lattice point inside of, sometimes you will get a pretty weird shaped cell. If you are more flexible, and if you decide to have more lattice points inside of your unit cell, you can get a better shape. For instance, if you use a cube with for lattice points, you can describe the Bravais lattice of a FCC very nicely. However, if you try to describe this very same lattice with a primitive cell, you will get a very weird thing.
Please, let me know if that helped you. I am also studying this subject, so we could discuss it together in more depth.

You can choose a Wigner-Seitz cell that is primitive and has the full symmetry of the lattice. I think one reason of using conventional unit cells is that it sometimes makes simpler the calculations.